Bottom-up rewriting for words and terms

For the whole class of linear term rewriting systems, we define \emph{bottom-up rewriting} which is a restriction of the usual notion of rewriting. We show that bottom-up rewriting effectively inverse-preserves recognizability and analyze the complexity of the underlying construction. The Bottom-Up class (BU) is, by definition, the set of linear systems for which every derivation can be replaced by a bottom-up derivation. Membership to BU turns out to be undecidable, we are thus lead to define more restricted classes: the classes SBU(k), k in N of Strongly Bottom-Up(k) systems for which we show that membership is decidable. We define the class of Strongly Bottom-Up systems by SBU = U_{k in \} SBU(k). We give a polynomial sufficient condition for a system to be in $\SBU$. The class SBU contains (strictly) several classes of systems which were already known to inverse preserve recognizability: the inverse left-basic semi-Thue systems (viewed as unary term rewriting systems), the linear growing term rewriting systems, the inverse Linear-Finite-Path-Ordering systems.